This Demonstration shows the data-fitting process to a three-parameter lognormal distribution. The built-in Mathematica function RandomVariate generates a dataset of pseudorandom observations from a lognormal distribution with "unknown" parameters , , and . You can use the sliders to propose values for these parameters and at the same time check the goodness-of-fit tests table, making sure that the -values indicate that there is a significant fit. If you select a location parameter that exceeds the minimum value of the pseudorandom dataset, an alarming message will appear. With "show parameters" selected, the unknown parameters are revealed in blue, as well as estimates of those parameters (see Details).

Snapshots

Details

During the fitting process, you can choose among four graphical displays: the cumulative distribution function (CDF) plot, the probability density function (PDF) plot, the quantile plot, and the density plot.

The "help" option reveals a table with the parameters , , and . The "empirical" parameters (blue) are locked by the "seed" slider and correspond to the generation process of the pseudorandom sample , . The "estimated" parameters (black) derive from the pivotal quantity and the Newton–Raphson technique, which are applied to estimate the location parameter , as well as from the built-in Mathematica functions EstimatedDistribution or FindDistributionParameters on the sample , to estimate the parameters and , using either the maximum likelihood or the method of moments.